Prove that if if $a < b < c$ and $f:[a,b] \to\mathbb{R}$ is integrable on $[a,b]$ and integrable on $[b,c]$ then $f$ is integrable on $[a,c]$.

This proof is in the context of Riemann intervals, and should rely on using a "partition". This is my attempt at a proof. Please tell me if it's decent or recommend alternate strategies.

Suppose $f:[a,b] \to\mathbb{R}$ is integrable on $[a,b]$ and integrable on $[b,c]$. By definition, there exists a partition $p_1$ of $[a,b]$ such that $p_1 = \{x_a, x_{a+1}, x_{a+2}...x_b\}$ and there exists a partition $p_2$ of $[a,b]$ such that $p_2 = \{x_b, x_{b+1}, x_{b+2}...x_c\}$. Let $p_3 = p_1 \cup p_2$.

I'm not sure how to make the final conclusion without it sounding like I'm just asserting it. Any help?

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  • $\begingroup$ You haven't used integrability anywhere. You haven't said anything about the partitions, just that they exist. What is the definition of integrable that you are using? $\endgroup$ – Michael Albanese Oct 22 '13 at 6:38
  • $\begingroup$ Good job, math.SE backend. This question is only three-and-a-third years old. $\endgroup$ – Patrick Stevens Mar 14 '17 at 22:38

$f:[a,c]\to \mathbb{R}$ is Riemann-integrable iff for every $\epsilon > 0$, there is a partition $\mathcal{P}$ of $[a,c]$ such that $$ U(f,\mathcal{P}) - L(f,\mathcal{P}) < \epsilon $$ where $U$ and $L$ denote the upper and lower sums respectively.

Now, if $f\in \mathcal{R}[a,b]$, then there is a partition $\mathcal{P}_1$ of $[a,b]$ such that $$ U(f,\mathcal{P}_1) - L(f,\mathcal{P}_1) < \epsilon $$ and a partition $\mathcal{P}_2$ of $[b,c]$ such that the similar inequality holds.

Now take $\mathcal{P} = \mathcal{P}_1\cup \mathcal{P}_2$, which is a partition of $[a,c]$. Now what can you say about $$ U(f,\mathcal{P}) - L(f,\mathcal{P})? $$ You need to write down the definition of each to find out how this relates to the inequalities involving the $\mathcal{P}_i$'s.

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